Robert Burns did not attend university but many of his family, friends and others of his acquaintance did.

 Listed below are the ones I know about who attended Glasgow University.

 Robert Burns Jnr. (1802) surviving twin of first twins to Jean Armour.
Thomas Wallace (1741) brother of Mrs Dunlop of Dunlop.
Patrick Miller (1743) landlord of Ellisland farm.
Alexander Hamilton (1756) brother of Gavin Hamilton.
James Dalrymple of Orangefield (1767) an early patron of Burns and cousin to Earl Glencairn.
William Auld (1728) “Daddy Auld” minister of Mauchline (1742-1791).
William Dalrymple MA (1740) “Dalrymple mild” of the The Kirks Alarm. Minister of Ayr (1746 -1814) who baptised Burns.
Clarinda’s (Agnes Craig) husband James M’Lehose (1767) her uncle John M’Laurin (1738) her cousins Alexander Craig and William Craig (1758).
Claud Alexander (1739) and his sons Alexander (1761), Claud (1765), and Boyd (1770) of Ballochmyle.
William McGill MA (1753) “Doctor Mac” of the Kirks Alarm.
Stephen Young MA (1770) “Barr Steenie” in The Kirks Alarm.
William McQuade MA (1756) of the “Twa Herds”.
Robert Duncan MA (1770) of the “Twa Herds”.
James Oliphant MA (1756) of “The Ordination”.
Alexander Fergusson (1763) of “The Whistle”.
Henry Erskine (1764) of “The Dean of Faculty”.
John McMath MA (1772) is in one of Burns’s political epistles.
John Wilson (1768) Dr. Hornbook (A hornbook is two slices of horn cleaned, flattened and polished with tracts between).
George Hamilton MA (1775) of a “Reply to a Reproof”.
James McKinlay (1773) of “The Ordination”.
William Tennant MA (1778) “Preacher Willie” in “Letter to James Tennant, Glenconner”.
James Steven (1777) of “The Calf”.
Richard Oswald (1785) husband of Lucy of the song “O What ye wha’s in yon toun?”
James Peebles MA (1801) son of “Peebles frae the Waterfit”.
John Russell (1799) and Alexander Russell (1800) sons of “Black Russell” of the “Holy Fair”.
John Tennant (1808) son of “Wabster Charlie”.
James Candlish MA (1819) and Robert Candlish MA (1823) sons of Jane Smith a “Mauchline Belle”.
William Tennant (1816) son of “Auchinbey”.
John Ronald (1825) of the “Ronalds of the Bennals”.